Semiregular Hermite tetrahedral finite elements. (Q1771832)

The authors examine tetrahedral Hermite \(C^0\)-elements satisfying the maximum angle condition. They introduce the notion ``a semiregular tetrahedron'', which is characterized as follows: There exists a constant \(\omega _0 < \pi \) such that the inner angle between its arbitrary two faces is less than or equal \(\omega _0\). Three basic manners, in which semiregular tetrahedra may degenerate, are described. Further, special degrees of freedom of semiregular tetrahedral Hermite elements are defined to get the best interpolation properties of these finite elements. Several new interpolation theorems are proved. They are applied to finite element approximation of the Poisson equation with mixed boundary conditions on a bounded polyhedral domain. The rate of convergence \(O(h^{3-\varepsilon })\) in \(H^1\)-norm is derived for any \(\varepsilon > 0\).